Improved pilot data aided feed forward based on maximum likelihood for carrier phase jitter recovery in coherent optical orthogonal frequency division multiplexing

Abstract

Pilot data aided feed forward (PAFF) carrier recovery is essential for phase noise tracking in coherent optical receivers. This paper describes a new PAFF system based on new pilot arrangement and maximum likelihood (ML) to estimate the phase jitter in coherent receiverinduced by local oscillator’s lasers and sampling clock errors. Square M-ary quadrature amplitude modulation (M-QAM) (4, 16, 64, and 256) schemes were used. A detailed mathematical description of the method was presented. The system performance was evaluated through numerical simulations and compared to those with noise-free receiver (ideal receiver) and feed forward without ML. The simulation results show that PAFF performs near the expected ideal phase recovery. Results clearly suggest that ML significantly improves the tolerance of phase error variance. From bit error rate (BER) sensibility evaluation, it was clearly observed that the new estimation method performs better with a 4-QAM (or quadrature phase shift keying (QPSK)) format compared to three others square QAM schemes. Analog to digital converter (ADC) resolution effect on the system performance was analyzed in terms of Q-factor. Finite resolution effect on 4-QAM is negligible while it negatively affects the system performance when M increases.

Key words： coherent optical orthogonal frequency division multiplexing (CO-OFDM); phase noise; feed forward (FF); maximum likelihood (ML); phase error variance; bit error rate (BER); Q-factor

Introduction

Coherent optical modems have been the main focus in optical industry in recent years because of increasing demands of data rate driven by the application of new technologies and the exponential growth of electronic processing speeds [ 1, 2]. Using multicarrier modulation technique like orthogonal frequency division multiplexing (OFDM) with coherent detection is more attractive for future optical transmission system. This combination of coherent detection and OFDM modulation improves the chromatic dispersion and polarization mode dispersion (PMD) estimation and mitigation. Secondly, the overlaps of OFDM subcarriers result in high optical spectral efficiency [ 3]. Moreover, when combined with square M-ary quadrature amplitude modulation (M-QAM) constellation, coherent optical orthogonal frequency division multiplexing (CO-OFDM) can meet the growing need for higher spectral efficiencies in future optical transport system [ 4]. However, one of the most severe impairment that affects coherent systems employing high-order modulation formats is the presence of phase noise and sampling clock errors [ 5, 6].

Besides phase noise and sampling errors, phase jitter induced by sampling clock errors in analog digital converter (ADC) and local oscillator (LO) lasers phase noise is considered to have markedly negative impact on coherent receivers as reported in previous literature [ 7]. Without digital signal processing (DSP) for phase recovery, the phase jitter in the receiver can destroy the orthogonality among the OFDM subcarriers [ 8, 9]. Hence, the phase jitter must be estimated and compensated, especially for optical OFDM receivers. So far, there are only a few reports on this issue which compensate individually sampling clock errors and lasers phase noise. One way is compensating the sampling clock errors by re-sampling in the time domain [ 10] and transmitting a dedicated clock signal [ 11, 12]. Re-sampling requires interpolation of sample points while transmitting a dedicated clock signal needs more circuits in the transmitter. Another way is the compensation of the lasers phase noise. For this issue, many algorithms have been introduced; such as common phase error (CPE) based [ 13] and radio frequency-pilot (RF-pilot) [ 14] based phase noise compensation which shows a superior tolerance to phase noise, compared to the first method, independent of the symbol length. These compensation methods mitigate either sampling errors or phase noise, which are two causes of phase jitter impairment.

In this paper, we comprehensively studied for the first time the carrier phase noise jitter (PNJ) as an impairment resulting from the combination of sampling clock errors and local oscillator phase noise at a CO-OFDM receiver. We assumed there was perfect chromatic dispersion compensation, which could be considered as a jitter source [ 8]. We evaluated the phase estimation based on a new system of diagonal pilot arrangement in the transmitter and a feed forward (FF) scheme which re-uses the pilot subcarriers in the receiver. Then the study was extended to maximum likelihood (ML) phase noise compensation systems employing arbitrary M-QAM constellations. A comparative analysis of feed forward maximum likelihood (FFML) with a pilot-data-aided algorithm using high square QAM modulation formats 4-QAM or quadrature phase shift keying (QPSK), 16-QAM, 64-QAM and 256-QAM was studied. This analysis was conducted at an Optisystem simulation platform in the presence of the random phase errors due to sampling errors in ADC, local oscillators’ lasers phase noise and additive channel noise. Except some additional DSP mathematical computation resources, the new phase jitter compensation method does not require additional hardware. This paper is divided into five sections including this introduction as Section 1. Section 2 describes the system architecture and mathematical model used for digital signal processing (DSP) in CO-OFDM. The main principles of new pilots’ arrangement in OFDM frame and FFML are detailed in Section 3. Section 4 highlights simulation conducted at an Optisystem platform to quantify the performance of the new estimation system. Section 5 provides a summary on the results of this study.

System architecture and mathematical model in CO-OFDM

System architecture

Figure 1 shows the CO-OFDM transmission system with DSP. The transmitter DSP helps to perform the following tasks: data conversion from serial to parallel, pulse shaping for symbols mapping onto M-QAM modulation format, inverse fast Fourier transform (IFFT) to convert signal from frequency to time domain, guard interval (GI) insertion to avoid inter symbol interference (ISI) and insertion of pilots for channel estimation at the receiver. Two ADC [ 15] are used to convert the time-domain digital signal to analog signal. A pair of electrical low-pass filters is used to remove the alias sideband signal. A radio-to-optical (RTO) up- converter transforms electrical signal into the optical domain using an inphase/quadrature (I/Q) optical modulator made by two Mach-Zehnder modulators (MZMs) and an 90° optical phase shifter.

Fig.1 Architecture of digital signal processing-based CO-OFDM

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At the receiver, the coherent OFDM optical-to-radio (OTR) front-end integrates local oscillator, balanced photo detectors to provide two signals corresponding to I and Q components filtered by low pass filters (LPF) [ 15]. These baseband signals are then sampled at their Nyquist rate by ADC where sampling errors occur. The DSP compensates for any impairment at the receiver and channel impairments (chromatic dispersion, polarization mode dispersion), and fiber non linear (NL) impairments.

This paper focuses on receiver impairment such as PNJ induced by LO lasers and sampling clock errors. Fiber channel impairments (chromatic dispersion and polarization dispersion) are beyond the scope of this study. We only considered the PNJ caused by the instantaneous phases of the transmitter LO lasers unlocked from each other, as well as circuit clock errors.

Mathematical description and phase jitter model

We have reported in previous research [ 16] that in the presence of PNJs after the fast Fourier transform (FFT) in coherent OFDM receiver, the received signal is expressed as

$$R_m\left(k\right)={\theta }_{\mathrm{0}}{H}_m\left(k\right)X_m\left(k\right)+I_m\left(k\right)+N_m,$$

(1)

where

$${\theta }_{\mathrm{0}}={\displaystyle \frac{\mathrm{1}}{N}}{\mathrm{e}}^{\mathrm{j}[{\omega }_{\mathrm{LD}}\left(t+ϵ\right)+{\phi }_{\mathrm{LD}}]}{\displaystyle \sum _{n=\mathrm{0}}^{N-\mathrm{1}}{\mathrm{e}}^{\frac{\mathrm{j2\pi }l{ϵ}_n}{T}}},$$

(2)

$$I_m={\displaystyle \sum _{k=\mathrm{0},k\ne m}^{N-\mathrm{1}}{H}_m\left(k\right)X_m\left(k\right)}{\displaystyle \sum _{n=\mathrm{0}}^{N-\mathrm{1}}{\theta }_{k-m}}{\mathrm{e}}^{\frac{\mathrm{j2\pi }}{N}\left(k-m\right)n},$$

(3)

where ${\theta }_{\mathrm{0}}$ is the optical OFDM transmitter/receiver phase noise, including optical and radio frequency (RF) local oscillator noise, and $X_m\left(k\right)$ is the transmitted baseband OFDM signal, $H_m\left(k\right)$ is overall end-to-end OFDM link impulse response function, $N_m$ is presumably white Gaussian noise which is the optical noise added from the optical amplifier throughout the optical fiber link. The fiber nonlinearity is not considered in this paper.

Considering the assumptions of free inter-symbol interference (ISI) and no fiber nonlinearities, Eq. (1) can be written as follows

$$R_m\left(k\right)={\theta }_{\mathrm{0}}{H}_m\left(k\right)X_m\left(k\right)+N_m.$$

(4)

From Eq. (2), it is noticed that transceiver and receiver oscillators phase noise contributions are located in the phase rotation term summing their individual phases. It should be noted that this is close to the real scenario since transceiver (Tx) and receiver (Rx) oscillators lasers cannot be located together because of chromatic dispersion. From this point on, the combined Tx and Rx lasers phase noise could be defined as

$${\theta }_{\mathrm{0}}\left(n{T}_{\mathrm{s}}\right)={\theta }_{\mathrm{Tx}}\left(n{T}_{\mathrm{s}}\right)+{\theta }_{\mathrm{Rx}}\left(n{T}_{\mathrm{s}}\right)+{\theta }_N\left(n{T}_{\mathrm{s}}\right),$$

(5)

where ${\theta }_N\left(n{T}_{\mathrm{s}}\right)$ is due to filtered noise.

It is assumed that frequencies of signal and LO lasers are synchronized so that the time variation in ${\theta }_N\left(n{T}_{\mathrm{s}}\right)$ is due to phase noise only, which can be modeled as a Wiener process [ 17].

$${\theta }_{\mathrm{0}}\left(k{T}_{\mathrm{s}}\right)\equiv {\theta }_k={\displaystyle \sum _{n=-\infty }^k{f}_n}.$$

(6)

Let’s assume at time $t=\mathrm{0}$ , phase ${\theta }_{\mathrm{0}}=\mathrm{0}$ . For a Wiener process, at any time $t=\mathrm{\Delta }t$ , the phase deviates from $\mathrm{0}$ , and the phase difference measured over $\mathrm{\Delta }t$ is a Gaussian random variable.

In Eq. (6), the increments $f_n$ are independent and identically distributed Gaussian random variables with zero mean and variance:

$${\sigma }_{\theta }^{\mathrm{2}}=\mathrm{2}\mathrm{\pi }\left(\mathrm{\Delta }f\cdot T_{\mathrm{s}}\right),$$

(7)

where $\mathrm{\Delta }f$ is the sum of the linewidth of transmitter laser and receiver local oscillator laser.

Principles of pilot data aided feed forward and feed forward maximum likelihood

Pilot data aided feed forward

Fig.2 Pilots arrangement for carrier phase jitter recovery (CPJR)

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In Fig.2, we denote by

Tab.1

$N_{\mathrm{s}}$	OFDM symbols numbers in one OFDM frame;
$N_{\mathrm{sb}}$	Number of subcarriers in one OFDM symbol;
$i$	OFDM index of subcarriers in the frame ( $i=\mathrm{1},\mathrm{2},\mathrm{...},N_{\mathrm{sb}}$ );
$k$	OFDM index of symbols in the frame ( $k=\mathrm{1},\mathrm{2},\mathrm{...},N_{\mathrm{s}}$ );
$X_{k,i}$	Position of subcarriers inside the frame;
$p_{k,i}$	Position of pilots inside the frame.

Pilots’ positions are defined as follows:

For $k\in \left[\mathrm{1}\mathrm{;}{N}_{\mathrm{s}}\right]$ and $i\in \left[\mathrm{1}\mathrm{;}{N}_{\mathrm{sb}}\right]$ , the pilots’ positions are given by the following mathematical expression:

$$p_{k,i}=p_{m{N}_{\mathrm{s}}+k,i}=\{\begin{array}{ll}{X}_{k,i},\hfill & i=k=\mathrm{2}m+\mathrm{1},\hfill \\ X_{k,N_{\mathrm{sb}}-k+\mathrm{1}},\hfill & k=\mathrm{2}m,i=N_{\mathrm{sb}}-k+\mathrm{1},\hfill \end{array}$$

where $m$ is an integer.

The main idea behind the novel pilot arrangement method is the insertion of pilots at transmitter in time-frequency optical OFDM frame space in the two congruent diagonals, such as shown in Fig. 2. Pilots are dispersed only in the two diagonals.

Such subcarriers allocation assigns the pilot subcarriers to all OFDM symbols across the entire signal spectrum, thus leading to highly accurate, interpolation-free phase estimation. The fact that one symbol contains a single pilot subcarrier, results in the same number of pilots in each frame as the number of data subcarriers within each symbol. This result is buffer-free flow. As the set of pilot is usually short, from a few data to a few symbols, a FF approach is suggested to speed up the required estimation process. The above mentioned features are totally different from those associated with block pilot insertion and comb pilot used in wireless communications.

Figure 3 illustrates the physical framing structure, where pilots are inserted for our carrier phase jitter estimation. One can see how pilots represent one symbol in a whole frame structure.

Fig.3 Physical framing structure of CO-OFDM signal with regularly inserted pilot symbol

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During the transmission, the pilot is impaired by the oscillators phase noise from both transmitter and receiver. The phase shift that occurs because of this impairment in a noise-free environment could be the same as the oscillators’ phase noise collected from data symbols. At the receiver, pilots are filtered out by low-pass filter and their phase is determined.

According to Eq. (5), we deducted the instantaneous pilot phase inserted in a symbol at the transceiver for estimation. It is written as

$${\theta }^{\left(p\right)}={\theta }_{\mathrm{Tx}}^p\left(t\right)+{\theta }_{\mathrm{Rx}}^p\left(t\right)+{\theta }_N^p\left(t\right).$$

(8)

Equation (5) is written with a discrete time index but a continuous time invariable t is employed throughout Eq. (8) simply because the discrete time processing at Nyquist rate is equivalent to continuous-time processing [ 18].

From Eq. (8), the receiver can easily estimate the pilot phase as ${\widehat{\theta }}^{\left(p\right)}$ , where the estimation error will depends on ${\theta }_N^p\left(t\right)$ . Then the receiver will de-rotate the received signal term in Eq. (4).

Principle with maximum likelihood

As we mentioned above, the phase noise is due to the linewidth difference between the transmitter and receiver lasers. The data pilot aided technique presented in this paper makes use of phase estimate carried out over the pilot set. As the symbol transmitted over the pilots is known, it is clear that the best estimator that can be used there is the ML estimator. For CO-OFDM, we assumed that $N_{\mathrm{p}}$ subcarriers are used as pilot subcarriers to estimate the phase noise. The ML of phase noise jitter is calculated as shown in Fig. 4. It calculates an estimate ${\widehat{\theta }}_{\mathrm{ML}}^{\left(\mathit{p}\right)}$ for the residual phase noise uncompensated by the pilot phase as follows:

$${\widehat{\theta }}_{\mathrm{ML}}^{\left(p\right)}=\arg \left\{{\displaystyle \sum _{k=\mathrm{0}}^{N_{\mathrm{p}}}{C}^{{\left(p\right)}^{\ast }}\left(k\right)S^{\left(p\right)}\left(k\right)}\right\}.$$

(9)

$$C^{{\left(p\right)}^{\ast }}={\displaystyle \frac{\mathrm{1}}{T_{\mathrm{s}}}}{\displaystyle {\int }_{\mathrm{0}}^{T_{\mathrm{s}}}r\left(t-\mathit{i}{T}_{\mathrm{s}}\right)S^{\ast }\left(k\right)\mathrm{d}t}$$

designates the conjugate of $C^{\left(p\right)}\left(k\right)$ which designates the pilot carrier symbols that are overhead (they do not carry any information content).

When a signal sample $S^{\left(p\right)}\left(k\right)$ corresponding to a pilot signal reaches the receiver, it is retrieved by the receiver and the above equation is computed sample by sample as illustrated in Fig. 4. Mathematically, this can be translated as the following equation

$$S\left(t\right)=c_{\mathrm{1}i}^{\mathrm{1}}{s}_{\mathrm{1}i}^{\mathrm{1}}\left(t-i{T}_{\mathrm{s}}\right)+c_{\mathrm{2}i}^{\mathrm{2}}{s}_{\mathrm{2}i}^{\mathrm{2}}\left(t-i{T}_{\mathrm{s}}\right)+c_{\mathrm{3}i}^{\mathrm{3}}{s}_{\mathrm{3}i}^{\mathrm{3}}\left(t-i{T}_{\mathrm{s}}\right)+\mathrm{...}+c_{ki}^k{s}_{ki}^k(t-i{T}_{\mathrm{s}})+\cdots +c_{Ni}^N{s}_{Ni}^N\left(t-i{T}_{\mathrm{s}}\right).$$

Then rewritten as

$$S\left(t\right)={\displaystyle \sum _{i=-\infty }^{+\infty }{\displaystyle \sum _{k=\mathrm{1}}^N{C}_{ki}{S}_k\left(t-\mathit{i}{T}_{\mathrm{s}}\right)},}$$

(10)

S(t) (which is denoted as X(k) in Eq. (1)) is the OFDM multi-carrier modulation transmitted signal.

Fig.4 Illustration of PA FFML for CPJR

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M-QAM constellations, simulation results

The pilot-data-aided feed forward carrier phase jitter recovery (PA FF CPJR) has been investigated for square m-ary QAM formats 4-QAM (or QPSK), 16-QAM, 64-QAM and 256-QAM. These square constellations are the easiest to generate [ 19] and are optimally immune against additive white Gaussian noise [ 20].

We compared three types of coherent OFDM receivers: a noise-free receiver (the ideal receiver), a receiver using PA FF CPJR without ML, and a receiver using PA FF CPJR with ML. Ideal receiver means a noise-free receiver in a simulation that knows the exact carrier phase achieved. In this kind of receiver, the penalty sensibility is caused only by quantization effects.

A computer simulation was used to verify the effectiveness of the PA FF carrier recovery employing ML in a CO-OFDM system at a 10 Gb/s transmission rate. OFDM system parameters used for the simulation are: number of subcarriers N = 512, FFT points= 1024, transmission link 1600 km standard single mode fiber (SSMF), sampling rate 1 GS/s and sampling period T_s = 1 ns, symbol period of 25.8 ns and guard time of 3.5 ns. All our simulation was based on each data point of 2000 OFDM symbols. The system is assumed to be free ISI.

M-QAM phase noise tolerance

PA FF with/without ML and a noise-free receiver (ideal receiver) are simulated to verify the performance of the new estimation method in terms of laser linewidth tolerance using square M-QAM schemes.

The sensibility penalty of optical signal to noise ratio (OSNR) as a function of phase error variance for 4-QAM, 16-QAM, 64-QAM and 256-QAM is shown respectively in Figs. 5(a)-5(d). It is shown that PA FFML performs much like a noise-free receiver and improves the linewidth tolerance when the phase error variance increases. Hence this demonstrates the effectiveness of the new estimation scheme for phase jitter recovery.

Fig.5 Penalty sensitivity for CO-OFDM noise-free receiver, PA FF CPJR receiver and PA FFML CPJR receiver. (a) 4-QAM (QPSK); (b) 16-QAM; (c) 64-QAM; (d) 256-QAM

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A quantitative performance of the new method in terms of linewidth tolerance is resumed in Table 1.

Tab.2 Linewidth requirements for PA FF CPJR with different square QAM constellations for OSNR= 1 dB

phase recovery techniques	4-QAM (QPSK)	16-QAM	64-QAM	256-QAM
max tolerable ${\sigma }_{\theta }^{\mathrm{2}}$ for 1 dB at BER= ${\mathrm{10}}^{-\mathrm{3}}$	max tolerable $\mathrm{\Delta }f$ for ${\displaystyle \frac{\mathrm{1}}{T_{\mathrm{s}}}}=\mathrm{1}\mathrm{Gbaud}$	max tolerable ${\sigma }_{\theta }^{\mathrm{2}}$ for 1 dB at BER= ${\mathrm{10}}^{-\mathrm{3}}$	max tolerable $\mathrm{\Delta }f$ for ${\displaystyle \frac{\mathrm{1}}{T_{\mathrm{s}}}}=\mathrm{1}\mathrm{Gbaud}$	max tolerable ${\sigma }_{\theta }^{\mathrm{2}}$ for 1 dB at BER= ${\mathrm{10}}^{-\mathrm{3}}$	max tolerable $\mathrm{\Delta }f$ for ${\displaystyle \frac{\mathrm{1}}{T_{\mathrm{s}}}}=\mathrm{1}\mathrm{Gbaud}$	max tolerable ${\sigma }_{\theta }^{\mathrm{2}}$ for 1 dB at BER= ${\mathrm{10}}^{-\mathrm{3}}$	max tolerable $\mathrm{\Delta }f$ for ${\displaystyle \frac{\mathrm{1}}{T_{\mathrm{s}}}}=\mathrm{1}\mathrm{Gbaud}$
ideal receiver	$\mathrm{56.55}\times {\mathrm{10}}^{-\mathrm{3}}$	9 MHz	$\mathrm{56.55}\times {\mathrm{10}}^{-\mathrm{4}}$	0.9 MHz	$\mathrm{53.40}\times {\mathrm{10}}^{-\mathrm{5}}$	85 kHz	$\mathrm{31.42}\times {\mathrm{10}}^{-\mathrm{5}}$	50 kHz
PA FF CPJR	$\mathrm{6.28}\times {\mathrm{10}}^{-\mathrm{4}}$	0.1 MHz	$\mathrm{3.14}\times {\mathrm{10}}^{-\mathrm{5}}$	5 kHz	$\mathrm{15.71}\times {\mathrm{10}}^{-\mathrm{5}}$	25 kHz	$\mathrm{50.27}\times {\mathrm{10}}^{-\mathrm{6}}$	8 kHz
PA FFML CPJR	$\mathrm{9.42}\times {\mathrm{10}}^{-\mathrm{3}}$	1.5 MHz	$\mathrm{6.28}\times {\mathrm{10}}^{-\mathrm{4}}$	0.1 MHz	$\mathrm{44}\times {\mathrm{10}}^{-\mathrm{5}}$	70 kHz	$\mathrm{44}\times {\mathrm{10}}^{-\mathrm{6}}$	7 kHz

Note: BER stand for bit error rate

For higher QAM constellations, the efficiency of the phase estimation reduces. This can be related to the fact that for lower OSNR other limiting factors like quantization and phase noise become more dominant.

Bit error rate sensitivity

In Fig. 6, we compared BER sensitivity versus OSNR for different QAM constellations using the CPJR with FFML.

Relation between theoretical expressions of OSNR and BER is given by [ 15]

$${\displaystyle \frac{E_{\mathrm{s}}}{N_{\mathrm{0}}}}=$$

$${\displaystyle \frac{E_{\mathrm{s}}}{N_{\mathrm{0}}}}={\displaystyle \frac{M-\mathrm{1}}{\mathrm{3}}}{\left(Q^{-\mathrm{1}}\left({\displaystyle \frac{{\log }_{\mathrm{2}}M}{\mathrm{2}}}{\left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\sqrt{M}}}\right)}^{-\mathrm{1}}\left(\mathrm{1}-\sqrt{\mathrm{1}-\mathrm{BER}}\right)\right)\right)}^{\mathrm{2}},$$

(11)

where, $E_{\mathrm{s}}/N_{\mathrm{0}}$ is the OSNR, M is the number of constellation points, BER is the target bit error rate and $Q\left(x\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathrm{erfc}\left({\displaystyle \frac{x}{\sqrt{\mathrm{2}}}}\right)$ is the Q function.

From Eq. (11), the BER is given by

$$\mathrm{BER}=\mathrm{1}-{\left(\mathrm{1}-{\displaystyle \frac{\mathrm{2}}{{\log }_{\mathrm{2}}M}}\left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\sqrt{M}}}\right)Q\left(\sqrt{{\displaystyle \frac{\mathrm{3}}{M-\mathrm{1}}}{\displaystyle \frac{E_{\mathrm{s}}}{N_{\mathrm{0}}}}}\right)\right)}^{\mathrm{2}}.$$

(12)

Figure 6 shows that 4-QAM attains a maximum BER of $\mathrm{2}\times {\mathrm{10}}^{-\mathrm{2}}$ . For 16-QAM, a maximum BER of $\mathrm{3}\times {\mathrm{10}}^{-\mathrm{2}}$ is attained. The maximum penalty for 64-QAM is $\mathrm{6}\times {\mathrm{10}}^{-\mathrm{2}}$ , while $\mathrm{8}\times {\mathrm{10}}^{-\mathrm{2}}$ BER is attained for 256-QAM.

Fig.6 BER performance comparison for different square M-QAM PA FF CPJR impacted by phase noise

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Impact of ADC resolutions on receiver performance

This part studies the effect of ADC resolutions on the system performance using PA FFML with different square M-QAM constellations. Figure 7 shows the Q-factor penalty versus different ADC resolutions.

From Fig. 7, it is noticed that there is a positive variation of Q factor ( $\mathrm{\Delta }Q=\mathrm{1.5}$ ) for 4-QAM, while 16-QAM, 64-QAM and 256-QAM attain respectively the following positive variations of Q factor: 2.5, 3.5 and 5.5. In other terms, 4-QAM is less sensible to big values of ADC resolutions compared to 16-QAM, 64-QAM and 256-QAM.

Fig.7 Impact of ADC resolutions on receiver sensitivity of CO-OFDM using square-QAM and PA FFML CPJR

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Conclusions

This paper demonstrated the use of PA FF CPJR in CO-OFDM employing square arbitrary M-QAM constellations as modulation formats. We introduced a new pilots arrangement in two congruent diagonals of OFDM frame leading to highly accurate, interpolation-free phase estimation. It is shown by mathematical deduction how the pilot’s phase is extracted at the receiver and used for phase estimation. The performance of the new method was demonstrated through simulations. The introduction of ML in PA FF was shown to be beneficial for the system. The phase error variance leading to 1-dB penalty was found to be $\mathrm{9.42}\times {\mathrm{10}}^{-\mathrm{3}}\mathrm{ra}{\mathrm{d}}^{\mathrm{2}}$ (4-QAM or QPSK), $\mathrm{6.28}\times {\mathrm{10}}^{-\mathrm{4}}\mathrm{ra}{\mathrm{d}}^{\mathrm{2}}$ (16-QAM), $\mathrm{44}\times {\mathrm{10}}^{-\mathrm{5}}\mathrm{ra}{\mathrm{d}}^{\mathrm{2}}$ (64-QAM) and $\mathrm{44}\times {\mathrm{10}}^{-\mathrm{6}}\mathrm{ra}{\mathrm{d}}^{\mathrm{2}}$ (256-QAM). BER versus OSNR simulations indicated that 4-QAM performs better. The impact of resolutions on system performance was studied. It was found that 4-QAM is immune to ADC resolutions. In contrast, the other three square QAM constellations performed poorly with small ADC resolution values and the effect of infinite ADC resolutions values was negligible.

In conclusion, simulations have shown that LO laser linewidth is involved in the quality of signal reception and 4-QAM outperforms 16-QAM, 64-QAM and 256-QAM. Moreover, with the new phase jitter estimation method, it is straightforward to interpolate the phase jitter with minimum mean square error at the receiver and thus the synchronization of CO-OFDM receiver can be achieved. This synchronization can improve the performance of CO-OFDM.

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